Robot Manipulator Redundancy Resolution

By Yunong Zhang and Long Jin

Introduces a revolutionary, quadratic-programming based approach to solving long-standing problems in motion planning and control of redundant manipulators 

This book describes a novel quadratic programming approach to solving redundancy resolutions problems with redundant manipulators. Known as ``QP-unified motion planning and control of redundant manipulators'' theory, it systematically solves difficult optimization problems of inequality-constrained motion planning and control of redundant manipulators that have plagued robotics engineers and systems designers for more than a quarter century.    

An example of redundancy resolution could involve a robotic limb with six joints, or degrees of freedom (DOFs), with which to position an object. As only five numbers are required to specify the position and orientation of the object, the robot can move with one remaining DOF through practically infinite poses while performing a specified task. In this case redundancy resolution refers to the process of choosing an optimal pose from among that infinite set. A critical issue in robotic systems control, the redundancy resolution problem has been widely studied for decades, and numerous solutions have been proposed. This book investigates various approaches to motion planning and control of redundant robot manipulators and describes the most successful strategy thus far developed for resolving redundancy resolution problems. 

• Provides a fully connected, systematic, methodological, consecutive, and easy approach to solving redundancy resolution problems
• Describes a new approach to the time-varying Jacobian matrix pseudoinversion, applied to the redundant-manipulator kinematic control
• Introduces The QP-based unification of robots' redundancy resolution
• Illustrates the effectiveness of the methods presented using a large number of computer simulation results based on PUMA560, PA10, and planar robot manipulators
• Provides technical details for all schemes and solvers presented, for readers to adopt and customize them for specific industrial applications 

Robot Manipulator Redundancy Resolution is must-reading for advanced undergraduates and graduate students of robotics, mechatronics, mechanical engineering, tracking control, neural dynamics/neural networks, numerical algorithms, computation and optimization, simulation and modelling, analog, and digital circuits. It is also a valuable working resource for practicing robotics engineers and systems designers and industrial researchers.

• Language: English
• Publisher: Wiley
• Release date: Sep 6, 2017
• ISBN: 9781119381433

Book preview

Table of Contents

Cover

Title Page

Copyright

Dedication

List of Figures

List of Tables

Preface

Acknowledgments

Acronyms

Part I: Pseudoinverse-Based ZD Approach

Chapter 1: Redundancy Resolution via Pseudoinverse and ZD Models

1.1 Introduction

1.2 Problem Formulation and ZD Models

1.3 ZD Applications to Different-Type Robot Manipulators

1.4 Chapter Summary

Part II: Inverse-Free Simple Approach

Chapter 2: G1 Type Scheme to JVL Inverse Kinematics

2.1 Introduction

2.2 Preliminaries and Related Work

2.3 Scheme Formulation

2.4 Computer Simulations

2.5 Physical Experiments

2.6 Chapter Summary

Chapter 3: D1G1 Type Scheme to JAL Inverse Kinematics

3.1 Introduction

3.2 Preliminaries and Related Work

3.3 Scheme Formulation

3.4 Computer Simulations

3.5 Chapter Summary

Chapter 4: Z1G1 Type Scheme to JAL Inverse Kinematics

4.1 Introduction

4.2 Problem Formulation and Z1G1 Type Scheme

4.3 Computer Simulations

4.4 Physical Experiments

4.5 Chapter Summary

Part III: QP Approach and Unification

Chapter 5: Redundancy Resolution via QP Approach and Unification

5.1 Introduction

5.2 Robotic Formulation

5.3 Handling Joint Physical Limits

5.4 Avoiding Obstacles

5.5 Various Performance Indices

5.6 Unified QP Formulation

5.7 Online QP Solutions

5.8 Computer Simulations

5.9 Chapter Summary

Part IV: Illustrative JVL QP Schemes and Performances

Chapter 6: Varying Joint-Velocity Limits Handled by QP

6.1 Introduction

6.2 Preliminaries and Problem Formulation

6.3 94LVI Assisted QP Solution

6.4 Computer Simulations and Physical Experiments

6.5 Chapter Summary

Chapter 7: Feedback-Aided Minimum Joint Motion

7.1 Introduction

7.2 Preliminaries and Problem Formulation

7.3 Computer Simulations and Physical Experiments

7.4 Chapter Summary

Chapter 8: QP Based Manipulator State Adjustment

8.1 Introduction

8.2 Preliminaries and Scheme Formulation

8.3 QP Solution and Control of Robot Manipulator

8.4 Computer Simulations and Comparisons

8.5 Physical Experiments

8.6 Chapter Summary

Part V: Self-Motion Planning

Chapter 9: QP-Based Self-Motion Planning

9.1 Introduction

9.2 Preliminaries and QP Formulation

9.3 LVIAPDNN Assisted QP Solution

9.4 PUMA560 Based Computer Simulations

9.5 PA10 Based Computer Simulations

9.6 Chapter Summary

Chapter 10: Pseudoinverse Method and Singularities Discussed

10.1 Introduction

10.2 Preliminaries and Scheme Formulation

10.3 LVIAPDNN Assisted QP Solution with Discussion

10.4 Computer Simulations

10.5 Chapter Summary

Appendix

Chapter 11: Self-Motion Planning with ZIV Constraint

11.1 Introduction

11.2 Preliminaries and Scheme Formulation

11.3 E47 Assisted QP Solution

11.4 Computer Simulations and Physical Experiments

11.5 Chapter Summary

Part VI: Manipulability Maximization

Chapter 12: Manipulability-Maximizing SMP Scheme

12.1 Introduction

12.2 Scheme Formulation

12.3 Computer Simulations and Physical Experiments

12.4 Chapter Summary

Chapter 13: Time-Varying Coefficient Aided MM Scheme

13.1 Introduction

13.2 Manipulability-Maximization with Time-Varying Coefficient

13.3 Computer Simulations and Physical Experiments

13.4 Chapter Summary

Part VII: Encoder Feedback and Joystick Control

Chapter 14: QP Based Encoder Feedback Control

14.1 Introduction

14.2 Preliminaries and Scheme Formulation

14.3 Computer Simulations

14.4 Physical Experiments

14.5 Chapter Summary

Chapter 15: QP Based Joystick Control

15.1 Introduction

15.2 Preliminaries and Hardware System

15.3 Scheme Formulation

15.4 Computer Simulations and Physical Experiments

15.5 Chapter Summary

References

Index

End User License Agreement

List of Illustrations

Chapter 1: Redundancy Resolution via Pseudoinverse and ZD Models

Figure 1.1 Block diagram of a kinematic-control system for a redundant robot manipulator by combining the MVN scheme (1.1) and ZD model, where c01-math-114 .

Figure 1.2 Geometry of a five-link planar robot manipulator used in simulations.

Figure 1.3 Joint-angle and joint-velocity profiles of a five-link planar robot manipulator synthesized by pseudoinverse-based MVN scheme (1.1) aided with TDTZD-U model (1.14).

Figure 1.4 (a) Motion process and (b) position error of a five-link planar robot manipulator synthesized by the pseudoinverse-based MVN scheme (1.1) and aided by the TDTZD-U model (1.14).

Figure 1.5 Position error of a five-link planar robot manipulator synthesized by a pseudoinverse-based MVN scheme (1.1) aided with the EDTZD-K model (1.9) or Newton iteration (1.15).

Figure 1.6 Position errors of a three-link planar robot manipulator synthesized by a pseudoinverse-based MVN scheme (1.1) aided with EDTZD-K model (1.9) with c01-math-141 .

Chapter 2: G1 Type Scheme to JVL Inverse Kinematics

Figure 2.1 (a) Motion process and (b) joint-angle profiles of a five-link redundant robot manipulator tracking the desired square path synthesized by the G1 type scheme (2.8).

Figure 2.2 (a) Desired path, actual trajectory, and (b) position error of a five-link redundant robot manipulator tracking square path synthesized by a G1 type scheme (2.8).

Figure 2.3 (a) Desired velocity and (b) velocity error of a five-link redundant robot manipulator tracking a square path synthesized by a G1 type scheme (2.8).

Figure 2.4 Motion process (a) and joint-angle profiles (b) of six-DOF redundant robot manipulator tracking the desired Z-shaped path synthesized by a G1 type scheme (2.8).

Figure 2.5 (a) Desired path, actual trajectory, and (b) position error of a six-DOF redundant robot manipulator tracking a Z-shaped path synthesized by a G1 type scheme (2.8).

Figure 2.6 (a) Desired velocity and (b) velocity error of a six-DOF redundant robot manipulator tracking a Z-shaped path synthesized by a G1 type scheme (2.8).

Figure 2.7 Hardware system (a) of six-DOF planar redundant robot manipulator with its structure platform (b). Source: Zhang et al. 2015. Reproduced from Y. Zhang, L. He, J. Ma et al., Inverse-free scheme of G1 type to velocity-level inverse kinematics of redundant robot manipulators, Figure 3, Proceedings of the Twelfth International Symposium on Neural Networks, pp. 99-108, 2015. ©Springer-Verlag 2015. With kind permission of Springer-Verlag (License Number 3978560065761).

Figure 2.8 Z-shaped-path-tracking experiment of six-DOF redundant robot manipulator synthesized by G1 type scheme (2.8) at joint-velocity level. Reproduced from Y. Zhang, L. He, J. Ma et al, Inverse-free scheme of G1 type to velocity-level inverse kinematics of redundant robot manipulators, Figure 4, Proceedings of the Twelfth International Symposium on Neural Networks, pp. 99-108, 2015. ©Springer-Verlag 2015. With kind permission of Springer-Verlag (License Number 3978560065761).

Chapter 3: D1G1 Type Scheme to JAL Inverse Kinematics

Figure 3.1 (a) Motion process, (b) desired path, and actual trajectory of a three-link redundant robot manipulator tracking a rhombus path synthesized by the D1G1 scheme (3.7) with c03-math-023 .

Figure 3.2 Joint-angle profiles (a) and position error (b) of a three-link redundant robot manipulator tracking a rhombus path synthesized by the D1G1 scheme (3.7) with c03-math-024 .

Figure 3.3 Joint-velocity profiles (a) and velocity error (b) of a three-link redundant robot manipulator tracking a rhombus path synthesized by the D1G1 scheme (3.7) with c03-math-025 .

Figure 3.4 (a) Joint-acceleration profiles and (b) acceleration error of a three-link redundant robot manipulator tracking a rhombus path synthesized by the D1G1 scheme (3.7) with c03-math-026 .

Figure 3.5 Position errors in rhombus-path-tracking task of a three-link robot manipulator synthesized by (a) pseudoinverse solution (3.4) and (b) inverse-free D1G1 scheme (3.7) with c03-math-027 .

Figure 3.6 (a) Motion process and (b) joint-angle profiles of a three-link redundant robot manipulator tracking a desired triangle path synthesized by a D1G1 scheme (3.7) with c03-math-076 .

Figure 3.7 Position errors in triangle-path-tracking task of the three-link robot manipulator synthesized by (a) pseudoinverse solution (3.4) and (b) inverse-free D1G1 scheme (3.7) with c03-math-077 .

Chapter 4: Z1G1 Type Scheme to JAL Inverse Kinematics

Figure 4.1 (a) Motion process and (b) joint-angle profiles of a three-link planar robot manipulator tracking a desired isosceles-trapezoid path synthesized by the Z1G1 type scheme (4.5).

Figure 4.2 (a) Joint-velocity profiles and (b) joint-acceleration profiles of three-link planar robot manipulator tracking isosceles-trapezoid path synthesized by Z1G1 type scheme (4.5).

Figure 4.3 (a) Motion process and (b) joint-angle profiles of a four-link planar robot manipulator tracking a desired isosceles-triangle path synthesized by the Z1G1 type scheme (4.5).

Figure 4.4 (a) Joint-velocity profiles and (b) joint-acceleration profiles of a four-link planar robot manipulator tracking an isosceles-triangle path synthesized by the Z1G1 type scheme (4.5).

Figure 4.5 Simulation results of a three-link planar robot manipulator tracking an isosceles-trapezoid path synthesized by the Z1G1 type scheme (4.5) with an initial end-effector position not on the desired path.

Figure 4.6 V-shaped-path-tracking experiment of a six-DOF redundant robot manipulator synthesized by the Z1G1 type scheme (4.5) at joint-acceleration level.

Chapter 5: Redundancy Resolution via QP Approach and Unification

Figure 5.1 Contradicting situations in equality-based collision-free formulation.

Figure 5.2 QP-based approach to redundancy resolution and torque control.

Figure 5.3 PUMA560 transients synthesized by a QP-based MTN scheme.

Figure 5.4 PUMA560 transients synthesized by a QP-based MKE scheme.

Figure 5.5 Comparison of (a) conventional and (b) presented MTN schemes.

Figure 5.6 Joint-torque profiles synthesized by other QP-based resolution schemes of (a) IIWT and (b) MAN.

Figure 5.7 Joint-torque profiles synthesized by other QP-based resolution schemes of (a ) MKE and (b) MVN.

Chapter 6: Varying Joint-Velocity Limits Handled by QP

Figure 6.1 (a) Hardware system and (b) model of a six-DOF planar robot manipulator.

Figure 6.2 Local configuration of a six-DOF planar robot manipulator.

Figure 6.3 Relationship between (a) c06-math-081 and c06-math-082 as well as (b) the relationship between c06-math-083 and c06-math-084 .

Figure 6.4 Relationship between (a) c06-math-085 and c06-math-086 as well as (b) the relationship between c06-math-087 and c06-math-088 .

Figure 6.5 Relationship between (a) c06-math-089 and c06-math-090 as well as (b) the relationship between c06-math-091 and c06-math-092 .

Figure 6.6 Desired line-segment path to be tracked by the end-effector of a six-DOF planar robot manipulator.

Figure 6.7 Snapshots for an actual task execution of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when a robot end-effector tracks a line-segment path. Reproduced from Z. Zhang, and Y. Zhang, Variable Joint-Velocity Limits of Redundant Robot Manipulators Handled by Quadratic Programming, Figure 5, IEEE/ASME Trans. Mechatronics, Vol. 18, No. 2, pp. 674-686, 2013.

Figure 6.8 Actual end-effector trajectory generated by a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme.

Figure 6.9 Simulated six-DOF planar robot manipulator and its joints' trajectories synthesized by a VJVL-constrained MVN scheme when the robot end-effector tracks a line-segment path.

Figure 6.10 (a) PPS c06-math-154 and (b) PPS c06-math-155 transmitted to joint motors when the end-effector tracks a line-segment path.

Figure 6.11 (a) PPS c06-math-156 and (b) PPS c06-math-157 transmitted to joint motors when the end-effector tracks a line-segment path.

Figure 6.12 (a) PPS c06-math-158 and (b) PPS c06-math-159 transmitted to joint motors when the end-effector tracks a line-segment path.

Figure 6.13 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when the robot end-effector tracks a line-segment path.

Figure 6.14 End-effector position error of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when the robot end-effector tracks a line-segment path.

Figure 6.15 Snapshots for actual task execution of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when the robot end-effector tracks an elliptical path.

Figure 6.16 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when the robot end-effector tracks an elliptical path.

Figure 6.17 End-effector position error of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when the robot end-effector tracks an elliptical path.

Figure 6.18 Joint-velocity profiles of (a) c06-math-218 and (b) c06-math-219 of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when its end-effector tracks a line-segment path much faster.

Figure 6.19 Joint-velocity profiles of (a) c06-math-220 and (b) c06-math-221 of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when its end-effector tracks a line-segment path much faster.

Figure 6.20 Joint-velocity profiles of (a) c06-math-222 and (b) c06-math-223 of a six-DOF planar robot manipulator synthesized by a the VJVL-constrained MVN scheme when its end-effector tracks a line-segment path much faster.

Figure 6.21 End-effector position error of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when its end-effector tracks a line-segment path much faster.

Figure 6.22 Joint-velocity profiles of a robot synthesized by a VJVL-constrained MVN scheme when its end-effector tracks an elliptical path much faster, which includes a profile of joint velocity c06-math-225 within and sometimes reaching limits of c06-math-226 .

Figure 6.23 End-effector position error of a six-DOF planar robot manipulator synthesized by a VJVL-constrained MVN scheme when its end-effector tracks an elliptical path much faster.

Figure 6.24 Iteration number and computing time of a numerical algorithm 94LVI (6.17) per sampling period in a much faster elliptical-path tracking task.

Chapter 7: Feedback-Aided Minimum Joint Motion

Figure 7.1 (a) Desired M-shaped path and (b) motion process of a six-DOF planar robot manipulator synthesized by a FAMJM scheme (7.23)–(7.25).

Figure 7.2 Maximum error variation tendency of a six-DOF planar robot manipulator synthesized by FAMJM scheme (7.23)–(7.25) when c07-math-112 increases from 0 to 100 during M-shaped-path-tracking execution.

Figure 7.3 Joint displacement variation tendency of a six-DOF planar robot manipulator synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks an M-shaped path.

Figure 7.4 Snapshots for actual task execution of a six-DOF planar robot manipulator synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks an M-shaped path.

Figure 7.5 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by FAMJM scheme (7.23)–(7.25) with c07-math-171 and c07-math-172 when a robot end-effector tracks an M-shaped path.

Figure 7.6 (a) Desired path, end-effector trajectory, and (b) position error for actual task execution of a six-DOF planar robot manipulator synthesized by FAMJM scheme (7.23)–(7.25) with c07-math-173 and c07-math-174 when a robot end-effector tracks an M-shaped path.

Figure 7.7 Snapshots of the actual task execution of a six-DOF planar robot manipulator synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a P-shaped path.

Figure 7.8 Joint-angle profiles of (a) c07-math-188 and (b) c07-math-189 synthesized by PBMJM scheme (7.27) when a robot end-effector tracks a larger M-shaped path.

Figure 7.9 Joint-angle profiles of (a) c07-math-190 and (b) c07-math-191 synthesized by PBMJM scheme (7.27) when a robot end-effector tracks a larger M-shaped path.

Figure 7.10 Joint-angle profiles of (a) c07-math-192 and (b) c07-math-193 synthesized by PBMJM scheme (7.27) when a robot end-effector tracks a larger M-shaped path.

Figure 7.11 Joint-angle profiles of (a) c07-math-194 and (b) c07-math-195 synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a larger M-shaped path.

Figure 7.12 Joint-angle profiles of (a) c07-math-196 and (b) c07-math-197 synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a larger M-shaped path.

Figure 7.13 Joint-angle profiles of (a) c07-math-198 and (b) c07-math-199 synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a larger M-shaped path.

Figure 7.14 Joint-angle profiles of (a) c07-math-200 and (b) c07-math-201 synthesized by PBMJM scheme (7.27) when robot end-effector tracks a larger P-shaped path.

Figure 7.15 Joint-angle profiles of (a) c07-math-202 and (b) c07-math-203 synthesized by PBMJM scheme (7.27) when a robot end-effector tracks a larger P-shaped path.

Figure 7.16 Joint-angle profiles of (a) c07-math-204 and (b) c07-math-205 synthesized by PBMJM scheme (7.27) when a robot end-effector tracks a larger P-shaped path.

Figure 7.17 Joint-angle profiles of (a) c07-math-206 and (b) c07-math-207 synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a larger P-shaped path.

Figure 7.18 Joint-angle profiles of (a) c07-math-208 and (b) c07-math-209 synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a larger P-shaped path.

Figure 7.19 Joint-angle profiles of (a) c07-math-210 and (b) c07-math-211 synthesized by FAMJM scheme (7.23)–(7.25) when a robot end-effector tracks a larger P-shaped path.

Chapter 8: QP Based Manipulator State Adjustment

Figure 8.1 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by state-adjustment scheme (8.1)–(8.3) without imposing a zero-initial-velocity constraint and with parameters c08-math-088 and c08-math-089 s.

Figure 8.2 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by state-adjustment scheme (8.1)–(8.3) without imposing a zero-initial-velocity constraint and with parameters c08-math-101 and c08-math-102 s.

Figure 8.3 (a) Motion process and (b) trajectories of a six-DOF planar robot manipulator synthesized by state-adjustment scheme (8.1)–(8.3) with a zero-initial-velocity constraint imposed (i.e., QP (8.13)–(8.14)) and with c08-math-149 and c08-math-150 s.

Figure 8.4 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by state-adjustment scheme (8.1)–(8.3) with zero-initial-velocity constraint imposed (i.e., QP (8.13)–(8.14)) and with c08-math-151 and c08-math-152 s.

Figure 8.5 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by state-adjustment scheme (8.1)–(8.3) with a zero-initial-velocity constraint imposed (i.e., QP (8.13)–(8.14)) and with c08-math-153 and c08-math-154 s.

Figure 8.6 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by state-adjustment scheme (8.1)–(8.3) with a zero-initial-velocity constraint imposed (i.e., QP (8.13)–(8.14)) and with c08-math-155 and c08-math-156 s.

Figure 8.7 (a) PPS c08-math-221 and (b) PPS c08-math-222 for controlling a six-DOF planar robot manipulator.

Figure 8.8 (a) PPS c08-math-223 and (b) PPS c08-math-224 for controlling a six-DOF planar robot manipulator.

Figure 8.9 (a) PPS c08-math-225 and (b) PPS c08-math-226 for controlling a six-DOF planar robot manipulator.

Figure 8.10 Motion transients of a physical six-DOF planar robot manipulator from initial state c08-math-227 to desired state c08-math-228 , which is synthesized by state-adjustment scheme (8.13)–(8.14) with c08-math-229 s.

Chapter 9: QP-Based Self-Motion Planning

Figure 9.1 Three-dimensional motion trajectories of a PUMA560 robot manipulator performing self-motion from configurations A to B with c09-math-124 .

Figure 9.2 (a) Joint-angle and (b) joint-velocity profiles of PUMA560 robot manipulator performing self-motion from configurations A to B with c09-math-125 .

Figure 9.3 Three-dimensional motion trajectories of a PUMA560 robot manipulator performing self-motion from configurations A to C with c09-math-173 .

Figure 9.4 (a) Joint-angle and (b) joint-velocity profiles of a PUMA560 robot manipulator performing self-motion from configurations A to C with c09-math-174 .

Figure 9.5 (a) Three-dimensional motion trajectories and (b) maximal end-effector position error of a PUMA560 robot manipulator from configurations E to F with c09-math-227 .

Figure 9.6 (a) Joint-angle and (b) joint-velocity profiles of a PUMA560 robot manipulator from configurations E to F with c09-math-228 .

Figure 9.7 (a) Joint-angle and (b) joint-velocity profiles of a PUMA560 robot manipulator performing self-motion from configurations E to F with c09-math-274 .

Figure 9.8 Three-dimensional motion trajectories of a PA10 manipulator performing self-motion (from c09-math-299 to c09-math-300 without moving the end-effector) with c09-math-301 .

Figure 9.9 (a) Joint-angle and (b) joint-velocity profiles of a PA10 manipulator performing self-motion (from c09-math-309 to c09-math-310 without moving end-effector) with c09-math-311 .

Figure 9.10 Resultant position error of a PA10 manipulator performing self-motion (from c09-math-312 to c09-math-313 theoretically without moving the end-effector) with c09-math-314 .

Figure 9.11 (a) Joint-angle and (b) joint-velocity profiles of a PA10 manipulator performing self-motion (from c09-math-362 to c09-math-363 without moving the end-effector) with c09-math-364 .

Figure 9.12 Three-dimensional motion trajectories of a PA10 manipulator performing self-motion (from c09-math-365 to c09-math-366 without moving the end-effector) with c09-math-367 .

Figure 9.13 Resultant position error of a PA10 manipulator performing self-motion (from c09-math-368 to c09-math-369 theoretically without moving the end-effector) with c09-math-370 .

Chapter 10: Pseudoinverse Method and Singularities Discussed

Figure 10.1 Block diagram of dynamic configuration of SMP for a redundant manipulator.

Figure 10.2 LVIAPDNN structure corresponding to dynamic equation (10.9).

Figure 10.3 Examples of three initial states of self-motion for a three-link redundant planar manipulator.

Figure 10.4 (a) Motion trajectories and (b) end-effector position error of a three-link robot performing self-motion with c010-math-135 .

Figure 10.5 (a) Joint-angle and (b) joint-velocity profiles of a three-link robot performing self-motion with c010-math-136 .

Figure 10.6 (a) Joint-velocity profiles and (b) motion trajectories of a three-link planar robot performing self-motion synthesized by the pseudoinverse-based method (10.12) with c010-math-184 and without considering constraint (10.8).

Figure 10.7 (a) Joint-velocity profiles and (b) motion trajectories of a three-link planar robot performing self-motion synthesized by the pseudoinverse-based method (10.12) with c010-math-185 and with constraint (10.8) imposed.

Figure 10.8 (a) Joint-velocity profiles and (b) motion trajectories of a three-link planar robot performing self-motion synthesized by LVIAPDNN solver (10.9) with c010-math-186 and with constraint (10.8) imposed as well.

Figure 10.9 (a) Motion trajectories and (b) end-effector position error of PUMA560 robot performing self-motion.

Figure 10.10 (a) Joint-angle and (b) joint-velocity profiles of a PUMA560 robot performing self-motion with c010-math-232 .

Figure 10.11 End-effector position errors of PUMA560 robot performing self-motion synthesized by LVIAPDNN solver (10.9) with different values of c010-math-248 (i.e., (a) c010-math-249 and (b) c010-math-250 ).

Figure 10.12 End-effector position errors of a PUMA560 robot performing self-motion synthesized by LVIAPDNN solver (10.9) with different values of c010-math-251 (i.e., (a) c010-math-252 and (b) c010-math-253 ).

Figure 10.13 (a) Motion trajectories and (b) end-effector position error of a PA10 robot performing self-motion with c010-math-256 .

Figure 10.14 (a) Joint-angle and (b) joint-velocity profiles of a PA10 robot performing self-motion with c010-math-257 .

Figure 10.15 (a) Motion trajectories and (b) c010-math-268 -proximity of a PA10 robot performing self-motion with c010-math-269 .

Chapter 11: Self-Motion Planning with ZIV Constraint

Figure 11.1 Safety device and limit-position indicators of a six-DOF planar robot manipulator.

Figure 11.2 Limits analysis of a six-DOF planar robot manipulator.

Figure 11.3 Block chart of zero-initial-velocity self-motion planning and control of a six-DOF planar robot manipulator.

Figure 11.4 Profiles of joint-velocity limits c011-math-104 , velocity-level angle-converted bounds c011-math-105 and corresponding bounds c011-math-106 with c011-math-107 of a six-DOF planar robot manipulator synthesized by QP (11.7)–(11.9) without imposing a zero-initial-velocity constraint.

Figure 11.5 Profiles of joint-velocity limits c011-math-108 , velocity-level angle-converted bounds c011-math-109 and corresponding bounds c011-math-110 with c011-math-111 of a six-DOF planar robot manipulator synthesized by QP (11.7)–(11.9) without imposing a zero-initial-velocity constraint.

Figure 11.6 Profiles of joint-velocity limits c011-math-112 , velocity-level angle-converted bounds c011-math-113 and final corresponding bounds c011-math-114 with c011-math-115 of a six-DOF planar robot manipulator synthesized by QP (11.10)–(11.12) with a zero-initial-velocity constraint imposed.

Figure 11.7 Profiles of joint-velocity limits c011-math-116 , velocity-level angle-converted bounds c011-math-117 , and final corresponding bounds c011-math-118 with c011-math-119 of a six-DOF planar robot manipulator synthesized by QP (11.10)–(11.12) with a zero-initial-velocity constraint imposed.

Figure 11.8 Joint-angle profiles of a six-DOF planar robot manipulator synthesized by self-motion schemes.

Figure 11.9 (a) PPS c011-math-129 , (b) PPS c011-math-130 , and (c) PPS c011-math-131 for controlling a six-DOF planar robot manipulator.

Figure 11.10 (a) PPS c011-math-132 , (b) PPS c011-math-133 , and (c) PPS c011-math-134 for controlling a six-DOF planar robot manipulator.

Figure 11.11 Self-motion task execution of a manipulator synthesized by the self-motion scheme (11.10)–(11.12).

Chapter 12: Manipulability-Maximizing SMP Scheme

Figure 12.1 Manipulability measures c012-math-090 synthesized by MMSMP scheme (12.2)–(12.5) and SMMVA scheme [203].

Figure 12.2 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by the MMSMP scheme (12.2)–(12.5).

Figure 12.3 Manipulability measures c012-math-093 of a six-DOF planar robot manipulator synthesized by the (a) MMSMP and (b) SMMVA schemes.

Figure 12.4 Self-motion task execution of a six-DOF planar robot manipulator synthesized by the MMSMP scheme (12.2)–(12.5). Source: IEEE 2012. Reproduced with permission of IEEE.

Chapter 13: Time-Varying Coefficient Aided MM Scheme

Figure 13.1 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator synthesized by a manipulability-maximizing scheme with a constant coefficient c013-math-136 .

Figure 13.2 (a) Motion trajectories as well as (b) end-effector path and trajectory of the end-effector of a six-DOF planar robot manipulator tracking an R path synthesized by time-varying coefficient aided manipulability-maximizing scheme (i.e., c013-math-139 ).

Figure 13.3 (a) Joint-angle and (b) joint-velocity profiles of a six-DOF planar robot manipulator tracking an R path synthesized by a time-varying coefficient aided manipulability-maximizing scheme (i.e., c013-math-140 ).

Figure 13.4 Comparison about manipulability index c013-math-155 synthesized by schemes with c013-math-156 , c013-math-157 (i.e., an MVN scheme) and c013-math-158 .

Figure 13.5 Joint-angle profiles (a) c013-math-164 and (b) c013-math-165 of corresponding limits of a six-DOF planar robot manipulator tracking an R path synthesized by TVCMM scheme with c013-math-166 .

Figure 13.6 Joint-angle profiles (a) c013-math-167 and